Decoherence of Majorana qubits by 1/f noise

Abstract

Qubits based on Majorana zero modes (MZMs) in superconductor-semiconductor nanowires have attracted intense interest due to claims that their error rates are suppressed exponentially with increasing nanowire length or decreasing temperature. However, here we show that these qubits are subject to substantial decoherence resulting from the high-frequency components of 1/f charge noise, which is ubiquitous in the materials surrounding the nanowire. This process excites quasiparticles in the bulk of the topological superconductor that cause qubit decoherence even under otherwise ideal conditions. Increasing nanowire capacitance suppresses this mechanism but exposes the qubits to decoherence from externally-generated quasiparticles. Therefore, achieving high-fidelity MZM qubits will require engineering strategies and compromises very similar to those needed for conventional superconducting qubits.

Figures (8)

• Figure 1: Quasiparticle generation in MZM nanowires by TLFs. A: Rendering of a superconducting-semiconducting nanowire device hosting MZMs. Atomic scale defects found in the materials surrounding the nanowire give rise to two-level fluctuators (TLFs) with different transition frequencies. B: Each TLF undergoes a series of sudden transitions between its states, each causing an instantaneous step in the chemical potential in the nanowire. C: The resulting frequency spectrum has components extending above the superconducting gap $\Delta$. A 1/f noise spectrum arises from an ensemble of two-level fluctuators (TLFs) Dutta:1981p497. D: (i) At zero temperature, bound Cooper pairs in the superconducting condensate are separated in energy from excited quasiparticles by twice the superconducting gap. (ii) Small sudden changes in chemical potential excite quasiparticle pairs from the superconducting condensate, (iii) which then travel to the ends of the wire and (iv) interact with the MZMs, changing the parity or (v) causing a qubit error.
• Figure 2: Excitation of quasiparticle pairs by a single two-level fluctuator (TLF) in a Kitaev chain.A: Sketch of the Hamiltonian of a Kitaev chain that hosts Majorana zero modes (MZMs). The parameters for the Kitaev chain (Eq. \ref{['eq:Kitaev']}), $w=350.8$, $\Delta=110$, $N \in \{159, 80, 48\}$ (corresponding to nanowire lengths of $\mathscr{L} \in \{10\;\unit{\um},~ 5\;\unit{\um}, 3\;\unit{\um} \}$), $\mu_1=0$, and $\mu_2=0.5657$ are determined from recent experiments aghaee_2025_nature_638_55 (see Supplementary Materials \ref{['supp_sec:nanowire_properties']} for further details). B: Probability of exciting at least one quasiparticle pair $P_{\rm QPP}$ versus time in the presence of a single TLF with switching rates $\Gamma$ of 200 and 20 as a function of time. C: Quasiparticle pair excitation probability after 5 ns, $P_{\rm QPP,~5ns}$, versus TLF transition rate $\Gamma$. The probability of exciting a quasiparticle pair in 100 ns grows with nanowire length (see c, inset), and it grows with $\Gamma$ until $\Gamma$ is of order $4\Delta/h$. The inset shows $P_{\rm QPP,~5ns}$ versus the nanowire length $\mathscr{L}$ for $\Gamma = 200$ and $20$. In B and C, $P_{\rm QPP,~5ns}$ is averaged over 20 different noise realizations. The error bars in C are the standard deviations of the means of subsets each containing 5 different noise realizations. These results demonstrate that the quasiparticle pair excitation rate is proportional to the chain length and increases with TLF switching rate $\Gamma$ up to a rate of $\sim200~{\rm GHz}$
• Figure 3: Decoherence of tetron qubit architecture caused by pairs of quasiparticles excited by 1/f noise.A: Schematic of a tetron qubit together with the even parity qubit states and odd parity leakage states. Because of the topological nature of MZMs, it is not possible for a single nanowire to have one empty and one full state. B: Decoherence by excited quasiparticles in the tetron qubit. Excited quasiparticles are mobile and have equal and opposite momenta. If one of the quasiparticles reaches an end of the nanowire and is absorbed by a MZM, there is a qubit leakage error. If both the quasiparticles in an excited pair go to opposite ends of the nanowire, there is a qubit phase (Z) error. C-D: Qubit dephasing due to bulk quasiparticle pairs for a TLF with rate $\Gamma$ that gives the maximum pair excitation rate, for a nanowire length $\mathscr{L} = 3$, as a function of (C) topological superconducting gap $\Delta$ and (D) inverse nanowire charge noise power $1/S_0$. These results were calculated using the Fermi Golden rule assuming that every excited quasiparticle pair causes a $Z$ error. The marked purple points denote the calculated dephasing time for current experimental parameters of the $3 \mu$m nanowire, using the Fermi Golden rule ($T_2^\ast \approx 0.3 \mu s$). E: Dependence of the inverse of the nanowire charge noise power $1/S_0$ on the nanowire capacitance $C_{\rm wire}$, where the marked purple point denotes the current experimental parameters ($C_{\rm wire} \approx C_{\rm dot}/5 = 2.25$fF and $S_0 = S_0^{\rm dot}/25 = 0.04 \mu$eV). These results demonstrate that increasing the capacitance of the topological nanowires is a promising avenue for decreasing the excitation of quasiparticle pairs by 1/f noise.
• Figure S1: Dependence of quasiparticle excitation rate on superconducting gap amplitude. Quasiparticle pair excitation rate $R_{\rm QPP}$ for a single TLF for superconducting gaps of $\Delta = 11~\mu\rm{eV}$ ($\Delta/h = 2.7~\rm{GHz}$), $\Delta = 36.7~\mu\rm{eV}$ ($\Delta/h = 8.9~\rm{GHz}$), $\Delta = 110~\mu\rm{eV}$ ($\Delta/h = 26.6$ GHz), $\Delta = 330~\mu\rm{eV}$ ($\Delta/h = 80$ GHz), and $\Delta = 990~\mu\rm{eV}$ ($\Delta/h = 239~\rm{GHz}$), with a chain length of $\mathscr{L} = 3\;\unit{\um}$, hopping amplitude $w = 350.8$ and with the TLF switching the chemical potential between the values $\mu_{1} = 0$ and $\mu_2 = 2.83$. The solid markers show $R_{\rm QPP} = P_{\rm QPP,~1ns}/(1\rm{ns})$ as calculated numerically for a Kitaev chain by first computing $P_{\rm QPP,~1ns}$, the probability of exciting at least one quasiparticle pair over $1~\rm ns$, averaged over 50 different realizations of the TLF. The dashed lines present $R_{\rm QPP}$ calculated using the Fermi golden rule as shown in Eq. \ref{['eq:fermi_gold_rule_result']}. In this figure, to make the calculations less time-intensive, we use the value $\delta \mu = 2.83$, which is a factor of $5$ larger than the value used in the main text. This plot demonstrates that the TLF transition rate at which the quasiparticle pair excitation rate varies systematically with $\Delta$ and is consistent with the linear dependence on $\Delta$ predicted by the Fermi golden rule.
• Figure S2: Dependence of rate of excitation of quasiparticle pairs (QPPs) on nanowire length. These calculations were done with a single two-level fluctuator (TLF) in Kitaev chains with lengths $\mathscr{L} = 3\;\unit{\um},~5\;\unit{\um},~10\;\unit{\um}$, hopping parameter $w = 350.8$, superconducting gap $\Delta = 110$, and one TLF switching between the chemical potential values $\mu_1 = 0$ and $\mu_2 = 0.5657$ at different transition rates $\Gamma$. A: Numerically calculated probability of exciting at least one QPP in a Kitaev chain over $5~\rm{ns}$, $P_{\rm QPP,~5~ns}^{\rm{num}}$ (solid markers with dashed-dot lines). The dashed lines show $P_{\rm QPP}^{\rm inco}$, the probabilities that would be obtained if the dynamics were completely incoherent and the QPP generation of successive transitions of the TLF were completely independent. B: Plot of the ratio $\mathscr{F} = P_{\rm QPP}^{\rm num}/P_{\rm QPP}^{\rm inco}$ (see Eq. \ref{['eqn:qpp_rate_mult_fact']}) versus TLF transition rate $\Gamma$. For the chains with lengths $\mathscr{L}=5~\mu$m and $\mathscr{L}=10~\mu$m, the ratio $P_{\rm QPP}^{\rm num}/P_{\rm QPP}^{\rm inco} \to 1$ as $\Gamma \to 0$. For the shortest chain length $\mathscr{L} = 3$ the MZM localization length is a large enough fraction of $\mathscr{L}$ to cause a noticeable deviation from $P_{\rm QPP}^{\rm inco} = 5~\rm{ns}\times \mathscr{L}(\delta \mu)^2/(\hbar v_{\rm F} \Delta)$ even at small $\Gamma$. C: Plot of $\mathscr{F}\Gamma$ scaled by $4 \Delta/h$. The dependence of $\mathscr{F}$ on nanowire length $\mathscr{L}$ is extremely weak. For the longer chains ($\mathscr{L} \in \{ 5\;\unit{\um},~10\;\unit{\um}\}$) in the regime where $R_{\rm{QPP}}$ is near its maximum, the multiplicative factor is $\mathscr{F}\Gamma \approx 0.7 \times 4\Delta/h$. This value is used in Eq. \ref{['eq:quasiparticle_excitation_rate_intermediate']} as well as in the main text to obtain the estimate of the rate of excitation of quasiparticle pairs.